Co-axial System of Circles
Category : JEE Main & Advanced
A system (or a family) of circles, every pair of which have the same radical axis, are called co-axial circles.
(1) The equation of a system of co-axial circles, when the equation of the radical axis and of one circle of the system are \[P\equiv lx+my+n=0\], \[S\equiv {{x}^{2}}+{{y}^{2}}+2gx+2fy+c=0\] respectively, is \[S+\lambda P=0\,\,\] \[(\lambda \] is an arbitrary constant).
(2) The equation of a co-axial system of circles, where the equation of any two circles of the system are
\[{{S}_{1}}\equiv {{x}^{2}}+{{y}^{2}}+2{{g}_{1}}x+2{{f}_{1}}y+{{c}_{1}}=0\]
and \[{{S}_{2}}\equiv {{x}^{2}}+{{y}^{2}}+2{{g}_{2}}x+2{{f}_{2}}y+{{c}_{2}}=0\]
Respectively, is \[{{S}_{1}}+\lambda \,({{S}_{1}}-{{S}_{2}})=0\]
or \[{{S}_{2}}+{{\lambda }_{1}}\,({{S}_{1}}-{{S}_{2}})=0\]
Other form \[{{S}_{1}}+\lambda {{S}_{2}}=0,\,\,\,\,\,(\lambda \ne -1)\]
(3) The equation of a system of co-axial circles in the simplest form is \[{{x}^{2}}+{{y}^{2}}+2gx+c=0\], where g is a variable and c is a constant.
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